Generating Functions of (Eventually) Constant Sequences

We find the (ordinary) generating function for constant sequences, and for sequences which are eventually constant (i.e. every term is = c after a certain point). After finding the generating function, we explore how our expression is connected to a more general result for non-constant sequences.

More on generating functions:

00:00 Generating functions
00:40 (i) Constant sequences
01:34 (ii) (a, c, c, c, ...)
02:46 (iii) Eventually constant sequences
06:36 Bigger picture

[rmschad]

This exact topic has been lurking in thr back of my mind for a couple weeks.

[haniamritdas]

This excellent presentation is very helpful, thank you.

[Jack_Callcott_AU]

Thanks for a useful and interesting video Dr Barker.

[peterhall]

As always, a very succinct insight.. Love your work.

[scottmiller]

This is interesting when applied to non-stationary time series, that is, time series whose statistics (such as the windowed standard deviation) change over time. Many (but not all) non-stationary time series become stationary in practice when differenced to a_k - a_{k-1} (so in the example of standard devation, the standard deviation of the difference series may be constant in practice).

[joefarrow]

What about for an exponential generating function? That'd be cool too

[islandfireballkill]

Weird stuff. You are forming a strange relationship between a sequence and a sequence's finite differences with this generating function thing and dividing by 1/(1-x).
It reminds me of z transforms somehow. Or at least discrete time fourier transforms and all the 1/(1-z)'s you get everywhere.

[wannabeactuary]

Hoping this is extended to partitions:-)

[alikaperdue]

A positive integer transformed by f(x) = (3x+lsb(x))/4, will eventually lead to a constant. It will be a power of two. Where lsb(x) is the least significant bit of x.
This is my personal feeling

[haniamritdas]

The logistic function appears as fundamental to number theory as the Pythagorean is to geometric analysis!

[MrRyanroberson]

2:54 if this is leading to = (2/3)/10^5 + 0.12345" then i actually wouldn't be surprised